138 ANALYTICAL MECHANICS 12. The time of descent along straight lines from a point on a vertical circle to the center and to the lowest point is the same. Find the position of the point. 13. A uniform cord of mass m and length I passes over a smooth peg and hangs vertically. If it slides freely, show that the tension of the cord equals 4 x ^ ~ x* ? when the length on one side is x. d 14. In an Atwood's machine experiment the sum of the two moving masses is m. Find their values if in t seconds they move through a distance h. 15. Given the height h of an inclined plane, show that its length must be ^r, in order that mi, descending vertically, shall draw w2 up the plane mji in the least possible time. 16. A gun points at a target suspended from a balloon. Show that if the target be dropped at the instant the gun is discharged, the bullet will hit the target if the latter is within the bullet's range. 17. Find the position where a particle sliding along the outside of a smooth vertical circle will leave the circle. 18. A particle falls towards a fixed point under the action of a force which equals yf~*3 where 7 is a constant and r is the distance of the particle from the fixed point. Show that starting from a distance a the particle 2 a^ will arrive at the fixed point with an infinite velocity in the time —7—• V37 19. A particle falls towards a fixed point under the attraction of a force which varies with some power of the distance of the particle from the center of attraction. Find the law of force, supposing the velocity acquired by the particle hi falling from an infinite distance to a distance a from the center to be equal to the velocity acquired in falling from rest from a distance a to a distance 7- 4 20. A particle is projected toward a center of attraction with a velocity equal to the velocity it would have acquired had it fallen from an infinite distance to the position of projection. Supposing the force of attraction to be yr~n, where 7 is a constant and r is the distance of the particle from the center of attraction, show that the time taken to cover the distance between the point of projection and the center of attraction is n+l .4 /n- + ./ n+!